466 Prof. Challis on the Nature of Aerial Vibrations. 

 and is plainly satisfied by the equation <I> = 0. Hence 



d<p d<p 



and <p is a function of z—a x t. Hence -j- is also a function of 



z— a x t t and a given state of the wave as to velocity is propa- 

 gated uniformly. Now it may be assumed that at some point 

 of the line of motion, viz. at the commencement of the series 

 of vibrations, the velocity and density vanish together ; so that 



p — 1 where --=0,and consequently, from what hasbeen shown 



above, where -j- = 0. Hence it appears from equation (2.) 

 that F(a,, b v c v t) = 0. Putting therefore f—l in that equa- 

 tion, it follows, since -~ and -£ are each functions of z—aJ. 

 at az 



that p is a function of the same quantity. Thus a given state 

 of the wave, both as to velocity and as to density, is propa- 

 gated with the same uniform velocity a v This result is in 

 perfect accordance with known facts respecting the transmis- 

 sion of articulate and musical sounds. 



I will take this occasion to advert again to a difficulty re- 

 garding the supposed effect of the development of heat on 

 aerial waves, which I have pointed out in the Philosophical 

 Magazine, vol. xxxii. pp. 283 and 498. Let the relation be- 

 tween the pressure and density, inclusive of the effect of tem- 

 perature, be expressed by the equation jp = a 2 p 1+/l ", as is allow- 

 able. Then putting 1+c for p, and supposing <r small, we 

 have 



= a 2 ( 1+ *) ^ (1+ k<r + &c. ). 



If now the terms k<r -f- &c. be neglected, the equation is of 

 the same form as that derived from the relation p = a' 2 p i and is 

 consistent with the uniform propagation of a given state of 

 velocity and density. But those terms stand in the way of 

 such a result ; and though they are of small amount, yet their 

 effect on the form of the wave is accumulative, and must in the 

 end entirely alter its character. This is the nature of the 

 difficulty I have alluded to, which, as we have seen, does not 

 exist in the case of ray-vibrations. 



Cambridge Observatory, 

 Nov. 21, 1848. 



